infinity in number and shape

Learning Objective:

• we are learning to understand what we mean by the infinite;
• we will discover some surprising arithmetical properties of infinity;
• we will discover that a surprising range of different sets of number have the same size
• we will discover with even greater surprise that there's more than one infinity and that some sets of numbers have a larger infinity than others
• we will explore well known paradoxes and look at infinite series

Investing:

• This is useful because we use infinite series to model and represent other functions and signals - although in practice we approximate these by stopping after a number of terms. This is the underlying mathematics behind digital music: CDs and mp3s represent the complicated waveforms of music by the addition of a finite approximation to the infinite series of sine waves that make up its shape.
• A functional (real-life) application is all the uses of calculus: although you may not have learnt this yet, differentiation (finding the steepness of graphs; the rate of change of a function) and integration (the area underneath a graph; the opposite of differentiation) depend on knowing that functions and series behave well in the limit to infinity.
• This skill leads to a deeper understanding of set theory, number, calculus, number theory, fractals and other interesting and useful advanced mathematics
• A 'free gift' with this skill is that we can now also think abstractly about the infinite - this may change how you think about life, faith and the universe!
• This could help you if you want to work in engineering, science, mathematics, economics, and a range of quantitative careers that use calculus on a regular basis.

Preparing:

• Are we ready? Do you have an inquisitive and flexible mind?
• You'll need to have a sound understanding of fractions, decimals and the like to get the best out some of the material later on, but gaps in your understanding of rational and irrrational numbers shouldn't stop you enjoying the earlier ideas.
• You will have a deeper understanding if you also know: a bit of set theory and notation, but we'll cover some of that on the way.

Discovering:

• As an introduction to the idea of the infinite you might enjoy:

• Try to predict the next one aloud or on a mini-whiteboard.
• Investigate...

Modeling:

• Here are some examples of people getting it right:
• Here are some examples of people getting it wrong in typical ways:
• Here are some more examples. Did they get it right or wrong? Explain how you know!

Discussing:

• What would this one be? Tell your learning partner. Convince them you're right.
• Explain how you know.
• How would you explain this to someone who was new to it?

Explaining:

• One way to do this is...
• Another approach might be...
• A useful shortcut is to...
• This works because...
• It doesn't work when...
• An exception is...
• Watch out for...
• A common mistake is...
• You can check your result by...
• We can prove this works by...

Practicing:

• Some straightforward examples.
• Some harder examples.
• Some mixed examples.
• Some non-examples to spot and some mixed questions with redundant, insufficient or contradictory data.
• You can demonstrate fluency by at least...

Sharing:

• A web page or wiki I have created to explain this can be found at...
• A presentation I have created and rehearsed looks like...
• A poster I have drawn or model I have made can be found...

Assessing:

• Check you've mastered this skill by...
• Show you understand by explaining...
• Prove you're an expert in... by...

Developing:

• Next we could learn...
• This leads to...
• Now try...